Are these statements about the performance of neural networks as a function of the number of hidden layers contradictory?
There are many problems requiring more than two hidden layers. Randomly select a recent Google journal paper on deep learning, you'll see their network could have something like 5 (or more) hidden layers.

Justin Domke wrote his notes for students, so he probably tried to make his points as simple as possible. For a "typical" machine learning problem that students would most likely work on, two hidden layers should be sufficient. But that doesn't add up for a real practical problem. "Deep" learning usually mean more than two hidden layers.

Number of hidden layers is network design that nobody knows for sure. Yoshua Bengio's suggestion is common and simple. It's not a mathmatic proof, but simply a guideline if you don't know what to  do. You just repeat and repeat, until you see the test error no longer improve.